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International Journal of Differential Equations
Volume 2015, Article ID 407930, 7 pages
http://dx.doi.org/10.1155/2015/407930
Research Article

Numerical Analysis of a Distributed Optimal Control Problem Governed by an Elliptic Variational Inequality

1Departamento de Matemática, EFB-FCEIA, Universidad Nacional de Rosario, Avenida Pellegrini 250, S2000BPT Rosario, Argentina
2Departamento de Matemática-CONICET, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina

Received 29 July 2015; Accepted 3 November 2015

Academic Editor: Kanishka Perera

Copyright © 2015 Mariela Olguín and Domingo A. Tarzia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energy . The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter goes to zero.